The former British prime minister, who left Downing Street widely unpopular, is back in favor with his party, Labour, which hopes his political skills can be an advantage as an election nears.A decade and a half after Tony Blair left Downing Street, one issue still defines the former British prime minister in the eyes of many Britons: his disastrous decision to join the U.S.-led invasion of Iraq.When Mr. Blair was given a knighthood by Queen Elizabeth II last year, more than a million people signed a petition demanding the honor be rescinded. And within his own Labour Party, he remained a complex figure, detested by those on the far left while grudgingly admired by some who noted that he was the party’s only leader to have won three consecutive British elections.Today, with the Labour opposition sensing rising power under the stewardship of its leader, Keir Starmer, Mr. Blair is suddenly, and rather remarkably, back in favor. For Mr. Starmer, embracing Mr. Blair sends a political message, underscoring Labour’s shift to the center. But the former prime minister also has charisma and communication skills that Mr. Starmer lacks, assets that could be useful as a general election approaches.Last month, the two men appeared onstage together, exchanging compliments at a glitzy conference organized by the Tony Blair Institute for Global Change — an organization that works for governments around the world, including autocratic ones, and churns out policies that could help Labour if it wins the next election.Mr. Blair, now 70, is graying, thinner and his face a little more gaunt than when he left Downing Street in 2007. But he still effortlessly held the stage as he told the audience that Britain would be in safe hands if Mr. Starmer won the next election.“It was like the apostolic succession was being declared,” said John McTernan, a political strategist and onetime aide to Mr. Blair, who added that “the chemistry between the two guys made you think they talk a lot and they understand each other.”Mr. Blair and Labour’s current leader, Keir Starmer, exchanged compliments onstage at a Tony Blair Institute conference.Stefan Rousseau/Press Association, via Getty ImagesJill Rutter, a former civil servant and a senior fellow at the Institute for Government, a London-based research institute, said Mr. Blair “has clearly been keen to reinsert himself as a big player in British politics,” but Mr. Starmer “is the first leader who seems prepared to let him do so.”The right-leaning Daily Telegraph newspaper was more blunt. “Tony Blair is preparing to rule Britain again — and Starmer might just let him,” read the headline of an opinion article.Mr. Blair led Labour into power in 1997 in a landslide victory and was prime minister for a decade, shifting the party to the center, helping to negotiate a peace deal in Northern Ireland and presiding over an economy strong enough to invest in health and education.But by the end of his tenure, and as Iraq descended into chaos, the public had soured on Mr. Blair, who, along with George W. Bush, the United States president, had justified the invasion with never-substantiated claims that the country had weapons of mass destruction. The invasion led to years of sectarian violence in Iraq and the rise of Islamist militant groups that became precursors to the Islamic State.Mr. Blair’s reputation post-Downing Street was further damaged by lucrative consultancy work for governments with dubious human rights records, seeming to confirm his affinity for wealth. Such questions have also been raised about his institute. London’s Sunday Times recently reported that the institute continued to advise the government of Saudi Arabia after the slaughter of the writer Jamal Khashoggi and still received money from the kingdom.The awarding of a knighthood to Mr. Blair last year prompted a street protest.Antony Jones/Getty ImagesIn a statement, the institute said, “Mr. Blair took the view then and is strongly of the view now — as he has said publicly — that whilst the murder of Mr. Khashoggi was a terrible crime that should never have happened, the program of social and economic change underway in Saudi Arabia is of immense and positive importance to the region and the world.”“The relationship with Saudi Arabia is of critical strategic importance to the West,” it added, and “therefore staying engaged there is justified.”None of these criticisms have stopped a rehabilitation that would have been inconceivable while Labour was led by Mr. Starmer’s predecessor, Jeremy Corbyn, a left-winger and a fierce political adversary of Mr. Blair’s. At the time, Mr. Starmer worked alongside Mr. Corbyn, and when Mr. Starmer became party leader in 2020, he initially kept Mr. Blair at arm’s length.Now, their ties are so warm that when the former prime minister recently celebrated his birthday at a London restaurant, Mr. Starmer dropped by to wish him well.“Tony has just kept going after a period in which it was almost like the Labour Party didn’t want him to be around,” said Alastair Campbell, Mr. Blair’s former spokesman. “I think people eventually think, ‘Say what you like about the guy, but he’s good at what he does; he’s still the most credible explainer of difficult situations.’”Some see a modern-day political parable in Mr. Blair’s return.“A lot of politics has now taken on the narrative of celebrity,” said Mr. McTernan, the political strategist, adding, “Tony, as a political celebrity, fell in the eyes of the public but he has earned his way back.”“It’s not about forgiveness about Iraq, but there is an arc of a narrative around Tony,” Mr. McTernan said, with Britons starting to “be ready to listen again.”Mr. Blair addressing British troops as prime minister in Basra, Iraq, in 2003.Pool photo by Stefan RousseauMr. Blair’s political rehabilitation has been helped by comparisons with a governing Conservative Party that has presided over political turmoil. Years of deadlock over Brexit were broken when Boris Johnson won a landslide election in 2019 — only to be driven out of Downing Street last year under a cloud of scandal. He was replaced by Liz Truss, the British prime minister with the shortest stint in history, before Rishi Sunak restored some stability.“We have had such a succession of failed prime ministers that, to look at someone who did command the stage, you do look back and say, ‘He was quite a big dominating prime minister,’” said Ms. Rutter.The institute’s output has also helped change Mr. Blair’s image, Mr. Campbell, his former spokesman, said. The former prime minister saw a gap for relatively nonideological research focusing on technocratic policymaking and tackling challenges such as artificial intelligence, digital policy and relations with the European Union.With about 800 staff members scattered around the world in Abu Dhabi, Accra, San Francisco, Singapore and New York, and a sleek, modern office in the West End of London, the institute has even had influence over the Conservative government, Ms. Rutter said, pointing to Mr. Blair’s proposal during the coronavirus pandemic to structure its vaccine program around giving as many people as possible a first shot.Mr. Campbell, his former spokesman, added that the work of the institute showed Mr. Blair in a new light, making money not just for himself but also “to build an organization, the fruits of which people are now seeing.”Perhaps the biggest question is: Now what?Mr. Blair, on the left of the second row, sat with other former prime ministers at the coronation of King Charles III this year.Pool photo by Richard Pohle“In the campaign, does an intervention from Tony help?” Mr. Campbell said of the coming election. “In my mind, it would; it would be big news. But that’s a tactical question.”If Labour wins power, more possibilities for influence would open up for Mr. Blair.Ms. Rutter suggests he has built up his institute in part because, when he was in Downing Street — which has relatively few staff members compared with government departments — he believed he had too few experts at his disposal.“The question is whether Blair is content to have an institute churning out reports that a Labour government may or may not want to look at, or will he be looking to be more of a power behind the throne,” she said.Mr. Blair, she added, “has tried to amass a huge piece of policy capability — the only problem for him now is that he’s not prime minister.” More